If n is a positive integer,and the coefficient of x^2 in the expansion of (1+x)^n + (1+2x)^n is 75,find the value(s) of n.
(HKCEE,A MATH 2002)
Thank you very much:D
Binomial Theorem一問
2012-03-20 1:44 am
回答 (1)
2012-03-20 2:05 am
✔ 最佳答案
Coefficient of x^2 term in (1 + x)^n is n(n - 1)/2!The x^2 term in (1 + 2x)^n is n(n - 1)/2![2x]^2, so the coefficient is 4n(n - 1)/2!
So n(n - 1)/2! + 4n(n - 1)/2! = 75
n^2 - n + 4n^2 - 4n = 150
5n^2 - 5n - 150 = 0
n^2 - n - 30 = 0
(n - 6)(n + 5) = 0
so n = 6 or - 5 (rej. since n is a positive integer).
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